Maths (Probabilities) Puzzle.
Discussion
Got a maths puzzle here I'm trying to solve, but I'm hopeless with probabilities.
Two players are playing a game with a coin.The coin is unbiased.
The game is turn based.
Player 1 - Tosses the coin. If it comes up heads Player 1 scores 1 point. If it comes up tails Player 1 scores 0 points.
Player 2 - Chooses an integer T and tosses the coin T times. If each toss is a head player 2 scores 2^(T-1) points. Otherwise player 2 scores 0.
Player 1 goes first.
Winner is first to 100 points.
Each time Player 2 takes a turn they choose a value for T which maximises their probability of winning the game.
Solving the chance of Player 2 winning seems simple if I knew how to determine each turn the value Player 2 should pick which gives the highest probability of winning. This is what I need help on
Anyone ?
Two players are playing a game with a coin.The coin is unbiased.
The game is turn based.
Player 1 - Tosses the coin. If it comes up heads Player 1 scores 1 point. If it comes up tails Player 1 scores 0 points.
Player 2 - Chooses an integer T and tosses the coin T times. If each toss is a head player 2 scores 2^(T-1) points. Otherwise player 2 scores 0.
Player 1 goes first.
Winner is first to 100 points.
Each time Player 2 takes a turn they choose a value for T which maximises their probability of winning the game.
Solving the chance of Player 2 winning seems simple if I knew how to determine each turn the value Player 2 should pick which gives the highest probability of winning. This is what I need help on
Anyone ?
Interesting...
Player 1 has obviously no bias, each turn can score 0 or 1 with 50% probability.
Player 2 can score an average of 0.5 each turn, same as player 1, regardless of T
So using a random value of T for player 2 will not bias the probability, and player 1 will win fractionally more than half the games, only because he goes first.
So player 2 must bias T depending on the actual scores at any time...
Probably worth working out probabilities for being player 2's turn with the scores at 99-99, then 99-98, 99-97, 98-98, and see if you can spot a pattern.
Player 1 has obviously no bias, each turn can score 0 or 1 with 50% probability.
Player 2 can score an average of 0.5 each turn, same as player 1, regardless of T
So using a random value of T for player 2 will not bias the probability, and player 1 will win fractionally more than half the games, only because he goes first.
So player 2 must bias T depending on the actual scores at any time...
Probably worth working out probabilities for being player 2's turn with the scores at 99-99, then 99-98, 99-97, 98-98, and see if you can spot a pattern.
P1 has a probability of 0.5 each turn to gain 1 point, so would take 200 turns to win.
If P2 sets T=8 each turn then they only need to hit it once to win (2^(8-1) = 128).
The probability of hitting 8 heads in a row is 0.5^8 = 0.00390625, thus the probability of them NOT hitting it each turn is 1-0.00390625 = 0.99609375.
The probability of not hitting it 200 times in a row is 0.99609375^200 = 0.45713347439222973069592389903499
If P2 sets T=8 each turn then they only need to hit it once to win (2^(8-1) = 128).
The probability of hitting 8 heads in a row is 0.5^8 = 0.00390625, thus the probability of them NOT hitting it each turn is 1-0.00390625 = 0.99609375.
The probability of not hitting it 200 times in a row is 0.99609375^200 = 0.45713347439222973069592389903499
JDMFanYo said:
If player 2 tosses the coin 1 million times, the probability is that they will be first to get 100 points, right?
The chance of tossing 1 million heads in a row is infinitesimally small.Edited by JDMFanYo on Tuesday 19th January 14:02
If Player 2 tosses it 1 million times and doesn't get 1 million heads they get 0 points.
They will lose 100-0.
Edited by him_over_there on Tuesday 19th January 14:09
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